Poorly embeddable metric spaces and Group Theory Mikhail Ostrovskii - PDF document

Poorly embeddable metric spaces and Group Theory Mikhail Ostrovskii St. John’s University Queens, New York City, NY e-mail: ostrovsm@stjohns.edu web page: http://facpub.stjohns.edu/ostrovsm March 2015, Geometric Group Theory on the Gulf Coast South Padre Island, Texas

• Introductory remarks: Group Theory is an important tool for constructing differ- ent exotic combinatorial and metric struc- tures. In my educational talk I plan to de- scribe constructions of metric spaces hav- ing poor embeddability properties into Ba- nach spaces, including basic tools used to prove poor embeddability and relevance of Group Theory. • Throughout the talk I shall provide refer- ences for those relevant results for which I do not plan to provide any details. Many of these references will be to my book “Met- ric Embeddings” (2013), which I shall cite as [ME]. Since the price of [ME] is rather large ( ≈ $100), I would like to mention that you can get a free preliminary version at Google Scholar. (There were some small changes and some theorem numbers etc can be slightly different).

• Motivation: Why do we want to embed metric spaces into Banach spaces? There are at least three reasons for this: • (1) In Combinatorial Optimization it was discovered that some problems, which were known to be computationally hard (the worst- case running time on an input of size n is known or believed to grow exponentially in n ) admit fast (in the sense of worst-case running time) approximation algorithms which are constructed using embeddings of cer- tain metric spaces into such Banach spaces as ℓ 2 (separable Hilbert space) and ℓ 1 (the space of absolutely summable sequences with the norm - sum of absolute values).

• What is an approximation algorithm? If we are looking, for example, for a largest set of pairwise adjacent vertices in a graph, a 1 2 -approximation algorithm is an algorithm which produces a set of pairwise adjacent vertices of size at least 1 2 of the maximal possible. • See [ME, Example 1.14 and Section 1.4] for more. See the books of Vazirani, Ap- proximation Algorithms and Williamson-Shmoys The Design of Approx- imation Algorithms for much more.

• (2) In Topology and K -theory it was pre- dicted by Gromov that some special cases of the Novikov and the Baum-Connes con- jectures can be proved using coarse em- beddings of the corresponding groups into sufficiently good Banach spaces. The first success on these lines is due to Guoliang Yu (2000), after that there were many other successes on these lines. • Do not ask me for more details on the Baum-Connes conjecture, since so far I have not found a source containing an under- standable for me reasonably detailed de- scription of it of a reasonable length.

• (3) In Group Theory, Guentner and Kaminker (2004) discovered that exactness of a group (Yu’s Property A) follows from sufficiently good embeddability of the group into a Hilbert space. Later this line of research was continued, see Nowak-Yu Large Scale Geometry , 2012, Section 5.9 and the cor- responding ‘Notes and Remarks’. • Plan of this lecture: Since we are inter- ested in ‘good’ embeddings it would be in- teresting to know what are the obstruc- tions to such embeddings. One of the main ways to get an obstruction is to prove a suitable Poincar´ e inequality. My lecture is devoted to Poincar´ e inequalities, their usage for establishing poor embeddability, and some ways of getting Poincar´ e inequal- ities using Group Theory. • We start by introducing those Banach and metric spaces which are important for this talk.

• Banach spaces of sequences: In all of the spaces below x i ∈ R , addition of se- quences and their multiplication by scalars are defined componentwise.  1 /p    ∞      { x i } ∞ i =1 : ||{ x i } ∞ | x i | p � ℓ p = i =1 || =  , < ∞     i =1 where 1 ≤ p < ∞ . � � { x i } ∞ i =1 : ||{ x i } ∞ ℓ ∞ = i =1 || = sup | x i | < ∞ . i ∈ N Observe that for p = 1 the formula for the ℓ p -norm becomes simpler: � ∞ i =1 | x i | . • Banach spaces of functions: L p (0 , 1), 1 ≤ p < ∞ is the space of those measurable functions on [0 , 1] for which the Lebesgue � 1 0 | f ( x ) | p dx is finite. Addition and integral scalar multiplication are defined pointwise, �� 1 � 1 /p . 0 | f ( x ) | p dx the norm is given by || f || = Usually we omit (0 , 1) from the notation of this space and denote it just L p .

• Reminder: Norm on a linear space X is a function ||·|| : X → R + (where R + is the set of all nonnegative real numbers) satisfying the conditions: – Triangle inequality: ∀ x, y ∈ X || x + y || ≤ || x || + || y || . – Symmetry: ∀ x ∈ X ∀ α ∈ R || αx || = | α | · || x || – If x � = 0, then || x || > 0. • The fact that the norms introduced above satisfy the triangle inequality requires non- trivial argument for 1 < p < ∞ .

• Metric spaces: A metric space is a set X endowed with a function d : X × X → R + (where R + is the set of all nonnegative real numbers) satisfying the conditions: – Triangle inequality: ∀ x, y, z ∈ X d ( x, z ) ≤ d ( x, y ) + d ( y, z ). – Symmetry: ∀ x, y ∈ X d ( x, y ) = d ( y, x ). – Separation axiom: ∀ x, y ∈ X x � = y ⇒ d ( x, y ) � = 0. – ∀ x ∈ X d ( x, x ) = 0. • The function d is called a metric on X .

• Graphs with graph distances: Let G = ( V ( G ) , E ( G )) be a graph, so V is a set of objects called vertices and E is some set of unordered pairs of vertices called edges. We denote an unordered pair consisting of vertices u and v by uv and say that u and v are ends of uv . • A walk in G is a finite sequence of the form W = v 0 , e 1 , v 1 , e 2 , . . . , e k , v k whose terms are alternately vertices and edges such that, for 1 ≤ i ≤ k , the edge e i has ends v i − 1 and v i . We say that W starts at v 0 and ends at v k , and that W is a v 0 v k -walk . The number k is called the lengths of the walk. A graph G is called connected if for each u, v ∈ V ( G ) there is a uv -walk in G . • If G is connected, we endow V ( G ) with the metric d G ( u, v ) = the length of the short- est uv -walk in G . The metric d G is called the graph distance . When we say “graph G with its graph distance” we mean the metric space ( V ( G ) , d G ).

– Groups with word metrics: Let G be a finitely generated group and S be a finite generating set of G . We assume that S does not contain the identity and is symmetric, that is, contains g if and only if it contains g − 1 . – The Cayley graph of G corresponding to the generating set S is the graph whose vertices are elements of G , ele- ments g 1 ∈ G and g 2 ∈ G are connected by an edge if and only if g − 1 1 g 2 ∈ S . – The graph distance of this graph is called the word metric because the distance between group elements g and h is the shortest representation of g − 1 h in terms of elements of S , such representations are called words in the alphabet S . • Each Banach space (and in particular all of the spaces introduced above) is considered as a metric space with the metric d ( x, y ) = || x − y || . The fact that the conditions for a metric space are satisfied by normed spaces follows from the definitions.

• Embeddings: By an embedding of a set X into Y we mean any (not necessarily in- jective or surjective) map of X into Y . • A map f : X → Y between two metric spaces is called an isometric embedding if it preserves distances, that is d Y ( f ( u ) , f ( v )) = d X ( u, v ) for all u, v ∈ X . • If there exists an isometric embedding of X into Y we say that X is isometric to a subset (subspace) of Y . • If an isometric embedding of X into Y is a bijection of X and Y , we say that X and Y are isometric . • The first observation which is worth men- tioning right away is that for some Banach spaces there are no geometric obstructions for embeddability (of course there are al- ways some trivial obstructions of the type: we cannot embed a metric space having large cardinality into a Banach space hav- ing small cardinality).

• Proposition [Fr´ echet (1910)]. Each count- able metric space admits an isometric em- bedding into ℓ ∞ (actually Fr´ echet proved this for a more general case of spaces ad- mitting countable dense set). – Proof: Let X = { u i } ∞ i =0 be a countable metric space. We introduce a map f : X → ℓ ∞ by f ( v ) = { d ( v, u i ) − d ( u i , u 0 ) } ∞ i =1 . Observe that || f ( v ) − f ( w ) || = sup | d ( v, u i ) − d ( w, u i ) | . i ∈ N – The triangle inequality implies sup | d ( v, u i ) − d ( w, u i ) | ≤ d ( v, w ) . i ∈ N – On the other hand, if v � = w , then at least one of v, w is among { u i } ∞ i =1 . Sup- pose that v ∈ { u i } ∞ i =1 . We get sup i ∈ N | d ( v, u i ) − d ( w, u i ) | ≥ | d ( v, v ) − d ( w, v ) | = d ( v, w ) . � . If the space is bounded, the subtracted term is not needed.